Design of Ultraspherical Window Functions with Prescribed Spectral Characteristics

Abstract

A method for the design of ultraspherical window functions that achieves prescribed spectral characteristics is proposed. The method comprises a collection of techniques that can be used to determine the three independent parameters of the ultraspherical window such that a specified ripple ratio and main-lobe width or null-to-null width along with a user-defined side-lobe pattern can be achieved. Other known two-parameter windows can achieve a specified ripple ratio and main-lobe width; however, their side-lobe pattern cannot be controlled as in the proposed method. A comparison with other windows has shown that a difference in performance exists between the ultraspherical and Kaiser windows, which depends critically on the required specifications. The paper also highlights some applications of the proposed method in the areas of digital beamforming and image processing

Exponential Window Family

“…The percentage error in the model changes between 0.065 and -0.086. This error range satisfies the error criterion in [11] which states that the predicted error in the normalized width must be smaller than 1 %.…”

“…The second design equation is the relation between the window length and the ripple ratio. To predict the window length for a given quantities R and w R , the normalized width parameter D w = 2w R (N-1) is used [11]. The relation between D w and R for the Exponential window with N = 51 and 101 is plotted in Figure 7.…”

“…In literature many windows have been proposed [4][5][6][7][8][9][10][11][12][13][14][15][16]. Since the best window depends on the applications, they are known as suboptimal solutions.…”

“…For beamforming applications, the higher sidelobe roll-off ratio means that it can reject far end interferences better [11]. For the design of nonrecursive digital filters, it reduces the far end attenuation for stopband energy [12], and reducing the energy leak from one band to another for speech processing [17].…”

Design of fir filters using exponential--Hamming window family

“…In the literature, many windows for signal spectrum and digital filter design applications have been * Correspondence: kavci1980@gmail.com proposed [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. According to the number of independent window parameters in their functions, they can be classified as fixed or adjustable [20].…”